Quantifying and understanding uncertainty plays a key role in reservoir management as the quality and robustness of business decisions are greatly impacted by the uncertainty estimates of the variables involved in the decision making process. Uncertainty arises due to scarcity of data, inaccuracy of measured data, and the inherent variability and heterogeneity in the geological formation and fluid properties. As a result, reservoir simulation model outputs, such as forecasted hydrocarbon production, may have significant uncertainty due to the variety of uncertain input parameters.
Monte Carlo simulation is an approach commonly used for uncertainty quantification. In Monte Carlo simulation, a large number of realizations of random inputs from a search space are generated and solved to obtain a set of model outputs, which can be further analyzed statistically. Monte Carlo simulation is conceptually straightforward and easy to implement, however, it requires a large computational effort. In particular, a large number of subsurface realizations and computationally intensive flow model simulations are typically needed to obtain statistically accurate results. Accordingly, Monte Carlo simulations are cost prohibitive in most real applications of reservoir simulation, especially for large scale problems.
Proxy models have been used as alternatives to Monte Carlo simulation for improving the efficiency of the traditional asset development workflows such as uncertainty assessments, history-matching and optimization of development plans. Proxy models are analytical functions calibrated to mimic the response of the numerical reservoir simulator, thus allowing for a fast evaluation of multiple combinations of input parameters. While proxy models are more computationally efficient compared to Monte Carlo simulation, they can require large amounts of training in order to accurately approximate the simulation responses. As the input parameter space increases, the amount of training needed for a proxy model diminishes the benefits obtained from the fast evaluations. Accordingly, because the cost of training grows exponentially with the parameter space dimension, applicability of proxy models has been limited to relatively small dimensional parameter spaces.
Furthermore, proxy models are typically constructed by statistical design, such as by using experimental design methods. A key disadvantage of experimental design methods is that they do not take into account the full probability distributions of the parameters consistently while creating the proxy model. In particular, the entire probability density functions (PDFs) are only used during post-processing and not used for sampling and design. Because all samples are equally weighted for proxy model generation, there is an inherent assumption that the distributions of these parameters are uniform. As a result, experimental design methods may not be appropriate when parameter distributions are arbitrary, which is common in petroleum applications. More particularly, linear regression techniques are the most popular methods to create analytical proxy models. However, the resulting proxies can be poor predictors of reservoir performance when strong non-linear effects occur.
Accordingly, a method is needed for reservoir evaluation that avoids the aforementioned shortcomings. In particular, a robust and efficient method is needed for generating proxy models over relatively large dimensional parameter spaces where non-linearities are common, such as when being applied to large scale reservoir simulation problems.